Strong edge coloring of Cayley graphs and some product graphs

A strong edge coloring of a graph \(G\) is a proper edge coloring of \(G\) such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper, we determine the exact value of the strong chromatic index of all unitary Cayley g...

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Bibliographic Details
Published inarXiv.org
Main Authors Suresh Dara, Mishra, Suchismita, Narayanan, Narayanan, Tuza, Zsolt
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.07.2021
Subjects
Cartesian coordinates
Color matching
Graph coloring
Graphs
Mathematics - Combinatorics
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Summary:A strong edge coloring of a graph \(G\) is a proper edge coloring of \(G\) such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper, we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 5 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by \(4\).
ISSN:2331-8422
DOI:10.48550/arxiv.2107.00718